= Solution
Use <structural induction> on the <propositional formula>. The induction claim is that its <truth value> under the all-one <Boolean valuation> is one. A variable has this <truth value> by definition. For the induction step, suppose the immediate subformulas $\alpha,\beta$ each have <truth value> one. Then
$$
\llbracket\alpha\lor\beta\rrbracket=1\lor1=1,\qquad
\llbracket\alpha\land\beta\rrbracket=1\land1=1,\qquad
\llbracket\alpha\to\beta\rrbracket=1\to1=1.
$$
These are all permitted construction cases. Thus every such <propositional formula> evaluates to one. Equivalently, the three <logical connectives> define <one-preserving Boolean functions>, and composition preserves this property. The variable-only base grammar matters: admitting a constant false formula would invalidate the claim.
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